Question: Determine how many solutions exist for the system of equations. ${8x-2y = -16}$ ${4x-y = -3}$
Explanation: Convert both equations to slope-intercept form: ${8x-2y = -16}$ $8x{-8x} - 2y = -16{-8x}$ $-2y = -16-8x$ $y = 8+4x$ ${y = 4x+8}$ ${4x-y = -3}$ $4x{-4x} - y = -3{-4x}$ $-y = -3-4x$ $y = 3+4x$ ${y = 4x+3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+8}$ ${y = 4x+3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.